In a stand mixer or large bowl, combine warm water, yeast, and sugar; stir until dissolved. Stir in 1/2 cup flour and let sit for 15 minutes, until mixture foams.
Add oil, salt, and 2 cups flour (reserving 1/2 cup). Mix with dough hook attachment or stir with a wooden spoon until a shaggy mass forms. If kneading with a dough hook, mix at low speed for 7 minutes, adding more of the reserved flour if the dough is too sticky. If kneading by hand, dust a clean surface with some of the reserved flour and knead until smooth and elastic, about 7 minutes, adding more of the reserved flour if the dough is too sticky. The dough should be soft and moist.
Place dough in a large clean bowl and cover with a clean dish towel. Leave in a warm place until dough has doubled in size, about 1 hour. Punch down dough, turn it out onto a lightly floured surface, divide it into eight pieces, and roll into balls. Cover with a towel and let rest for 10 minutes.
Preheat a large cast-iron skillet over medium heat. Working one at a time, roll each round into a 1/4-inch-thick circle that is about 8 inches in diameter, sprinkling dough with extra flour if it starts to stick.
Cook each pita one at a time in the skillet until an air pocket balloons, then flip and cook 1 minute more. Cover baked pitas with a clean dish towel to keep warm.